Commutative Algebra and Noncommutative Algebraic Geometry: Volume 1, Expository Articles

462 Pages·2015·3.93 MB·English

by  David Eisenbud

Inthe2012–13academicyear,theMathematicalSciencesResearchInstitute,Berke-
ley,hostedprogramsinCommutativeAlgebra(Fall2012andSpring2013)andNon-
commutativeAlgebraicGeometry and RepresentationTheory (Spring2013).There
havebeenmanysignificantdevelopmentsinthesefieldsinrecentyears;whatismore,
the boundary between them has become increasingly blurred. This was apparent
during the MSRI program, where there were a number of joint seminars on sub-
jects of common interest: birational geometry, D-modules, invariant theory, matrix
factorizations,noncommutativeresolutions,singularitycategories,supportvarieties,
andtiltingtheory,tonameafew.Thesevolumesreflectthelivelyinteractionbetween
thesubjectswitnessedatMSRI.
TheIntroductoryWorkshopsandConnectionsforWomenWorkshopsforthetwo
programsincludedlectureseriesbyexpertsinthefield.Thevolumesincludeanum-
ber of survey articles based on these lectures, along with expository articles and
researchpapersbyparticipantsoftheprograms.

Mathematical Sciences Research Institute
Publications
67
Commutative Algebra and
Noncommutative Algebraic Geometry
Volume I: Expository Articles
Mathematical Sciences Research Institute Publications
1 Freed/Uhlenbeck:InstantonsandFour-Manifolds,secondedition
2 Chern(ed.):SeminaronNonlinearPartialDifferentialEquations
3 Lepowsky/Mandelstam/Singer(eds.):VertexOperatorsinMathematicsandPhysics
4 Kac(ed.):InfiniteDimensionalGroupswithApplications
5 Blackadar:K-TheoryforOperatorAlgebras,secondedition
6 Moore(ed.):GroupRepresentations,ErgodicTheory,OperatorAlgebras,andMathematicalPhysics
7 Chorin/Majda(eds.):WaveMotion:Theory,Modelling,andComputation
8 Gersten(ed.):EssaysinGroupTheory
9 Moore/Schochet:GlobalAnalysisonFoliatedSpaces,secondedition
10–11 Drasin/Earle/Gehring/Kra/Marden(eds.):HolomorphicFunctionsandModuli
12–13 Ni/Peletier/Serrin(eds.):NonlinearDiffusionEquationsandTheirEquilibriumStates
14 Goodman/delaHarpe/Jones:CoxeterGraphsandTowersofAlgebras
15 Hochster/Huneke/Sally(eds.):CommutativeAlgebra
16 Ihara/Ribet/Serre(eds.):GaloisGroupsoverQ
17 Concus/Finn/Hoffman(eds.):GeometricAnalysisandComputerGraphics
18 Bryant/Chern/Gardner/Goldschmidt/Griffiths:ExteriorDifferentialSystems
19 Alperin(ed.):ArborealGroupTheory
20 Dazord/Weinstein(eds.):SymplecticGeometry,Groupoids,andIntegrableSystems
21 Moschovakis(ed.):LogicfromComputerScience
22 Ratiu(ed.):TheGeometryofHamiltonianSystems
23 Baumslag/Miller(eds.):AlgorithmsandClassificationinCombinatorialGroupTheory
24 Montgomery/Small(eds.):NoncommutativeRings
25 Akbulut/King:TopologyofRealAlgebraicSets
26 Judah/Just/Woodin(eds.):SetTheoryoftheContinuum
27 Carlsson/Cohen/Hsiang/Jones(eds.):AlgebraicTopologyandItsApplications
28 Clemens/Kolla´r(eds.):CurrentTopicsinComplexAlgebraicGeometry
29 Nowakowski(ed.):GamesofNoChance
30 Grove/Petersen(eds.):ComparisonGeometry
31 Levy(ed.):FlavorsofGeometry
32 Cecil/Chern(eds.):TightandTautSubmanifolds
33 Axler/McCarthy/Sarason(eds.):HolomorphicSpaces
34 Ball/Milman(eds.):ConvexGeometricAnalysis
35 Levy(ed.):TheEightfoldWay
36 Gavosto/Krantz/McCallum(eds.):ContemporaryIssuesinMathematicsEducation
37 Schneider/Siu(eds.):SeveralComplexVariables
38 Billera/Bjo¨rner/Green/Simion/Stanley(eds.):NewPerspectivesinGeometricCombinatorics
39 Haskell/Pillay/Steinhorn(eds.):ModelTheory,Algebra,andGeometry
40 Bleher/Its(eds.):RandomMatrixModelsandTheirApplications
41 Schneps(ed.):GaloisGroupsandFundamentalGroups
42 Nowakowski(ed.):MoreGamesofNoChance
43 Montgomery/Schneider(eds.):NewDirectionsinHopfAlgebras
44 Buhler/Stevenhagen(eds.):AlgorithmicNumberTheory:Lattices,NumberFields,
CurvesandCryptography
45 Jensen/Ledet/Yui:GenericPolynomials:ConstructiveAspectsoftheInverseGaloisProblem
46 Rockmore/Healy(eds.):ModernSignalProcessing
47 Uhlmann(ed.):InsideOut:InverseProblemsandApplications
48 Gross/Kotiuga:ElectromagneticTheoryandComputation:ATopologicalApproach
49 Darmon/Zhang(eds.):HeegnerPointsandRankinL-Series
50 Bao/Bryant/Chern/Shen(eds.):ASamplerofRiemann–FinslerGeometry
51 Avramov/Green/Huneke/Smith/Sturmfels(eds.):TrendsinCommutativeAlgebra
52 Goodman/Pach/Welzl(eds.):CombinatorialandComputationalGeometry
53 Schoenfeld(ed.):AssessingMathematicalProficiency
54 Hasselblatt(ed.):Dynamics,ErgodicTheory,andGeometry
55 Pinsky/Birnir(eds.):Probability,GeometryandIntegrableSystems
56 Albert/Nowakowski(eds.):GamesofNoChance3
57 Kirsten/Williams(eds.):AWindowintoZetaandModularPhysics
58 Friedman/Hunsicker/Libgober/Maxim(eds.):TopologyofStratifiedSpaces
c
59 Caporaso/M Kernan/Mustat,a˘/Popa(eds.):CurrentDevelopmentsinAlgebraicGeometry
60 Uhlmann(ed.):InverseProblemsandApplications:InsideOutII
61 Breuillard/Oh(eds.):ThinGroupsandSuperstrongApproximation
62 Eguchi/Eliashberg/Maeda(eds.):Symplectic,Poisson,andNoncommutativeGeometry
63 Nowakowski(ed.):GamesofNoChance4
65 Deift/Forrester(eds.):RandomMatrixTheory,InteractingParticleSystems,andIntegrableSystems
67–68 Eisenbud/Iyengar/Singh/Stafford/VandenBergh(eds.):CommutativeAlgebraandNoncommutative
AlgebraicGeometry
Commutative Algebra
and Noncommutative
Algebraic Geometry
Volume I: Expository Articles
Editedby
David Eisenbud
UniversityofCalifornia,Berkeley
andMathematicalSciencesResearchInstitute
Srikanth B. Iyengar
UniversityofUtah
Anurag K. Singh
UniversityofUtah
J. Toby Stafford
UniversityofManchester
Michel Van den Bergh
FondsWetenschappelijkOnderzoek-Vlaanderen
andUniversiteitHasselt
DavidEisenbud SrikanthB.Iyengar AnuragK.Singh
[email protected] [email protected] [email protected]
J.TobyStafford MichelVandenBergh
[email protected] [email protected]
SilvioLevy(SeriesEditor)
MathematicalSciencesResearchInstitute
[email protected]
TheMathematicalSciencesResearchInstitutewishestoacknowledgesupportbythe
NationalScienceFoundationforthepublicationofthisseries.
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitofeducation,
learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107065628
(cid:13)c MathematicalSciencesResearchInstitute2015
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2015
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
ISBN978-1-107-06562-8Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsfor
externalorthird-partyInternetwebsitesreferredtointhispublicationanddoesnotguarantee
thatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
CommutativeAlgebraandNoncommutativeAlgebraicGeometry,I
MSRIPublications
Volume67,2015
Contents
Preface ix
Growthfunctions 1
JASONP.BELL
Syzygies,finitelengthmodules,andrandomcurves 25
CHRISTINEBERKESCHANDFRANK-OLAFSCHREYER
Vectorbundlesandidealclosureoperations 53
HOLGERBRENNER
Heckealgebrasandsymplecticreflectionalgebras 95
MARIACHLOUVERAKI
Limitsincommutativealgebraandalgebraicgeometry 141
STEVENDALECUTKOSKY
Introductiontouniformityincommutativealgebra 163
CRAIGHUNEKEANDCLAUDIURAICU
Noncommutativemotivesandtheirapplications 191
MATILDEMARCOLLIANDGONÇALOTABUADA
Infinitegradedfreeresolutions 215
JASONMCCULLOUGHANDIRENAPEEVA
Poincaré–Birkhoff–Witttheorems 259
ANNEV.SHEPLERANDSARAHWITHERSPOON
Frobeniussplittingincommutativealgebra 291
KARENE.SMITHANDWENLIANGZHANG
FromBriançon–SkodatoScherk–Varchenko 347
DUCOVANSTRATEN
Theinterplayofalgebraandgeometryinthesettingofregularalgebras 371
MICHAELAVANCLIFF
Surveyonthe D-module fs 391
ULIWALTHER,WITHANAPPENDIXBYANTONLEYKIN
Introductiontoderivedcategories 431
AMNONYEKUTIELI
vii

CommutativeAlgebraandNoncommutativeAlgebraicGeometry,I
MSRIPublications
Volume67,2015
Preface
In the 2012–13academic year, the Mathematical SciencesResearch Institute,
in Berkeley, hosted programs in Commutative Algebra (Fall 2012 and Spring
2013) and Noncommutative Algebraic Geometry and Representation Theory
(Spring2013). Theprogramshad 174participantsvisitingforperiodsranging
between one and nine months, and many others for shorter periods and for
week-longworkshops.
There have been many significant developments in these fields in recent
years;whatismore,theonceratherstrictboundarybetweenthemhasbecome
increasinglyblurred. ThiswasapparentduringtheMSRIprogram, wherethere
were a number of joint seminars on subjects of common interest: birational
geometry, D-modules, invariant theory, matrix factorizations, non-commutative
resolution of singularities, singularity categories, support varieties, and tilting
theory, to name a few. This volume is intended to reflect, and stimulate, the
livelyinteractionbetweenthetwosubjectsthatwewitnessedatMSRI.
TheIntroductoryWorkshops andConnectionsforWomen Workshopsforthe
twoprogramsincludedlectureseriesbyexpertsinthefield;thevolumeincludes
anumberofsurveyarticlesbasedontheselectures. Therearealsoexpository
articlesandresearchpapersbysomeoftheotherparticipantsoftheprograms.
Inadditiontotheeditorsofthisvolume,theorganizersoftheprogramsandthe
IntroductoryandConnectionsforWomenworkshopswereMikeArtin,Georgia
Benkart,VictorGinzburg,BernardKeller,EllenKirkman,EzraMiller,Claudia
Polini,IdunReiten,SueSierra,KarenE.Smith,CatharinaStroppel,Alexander
Vainshtein,LaurenWilliams,andEfimZelmanov. Wetakethisopportunityto
expressourthankstotheparticipants,ourco-organizers,theMSRIstaff,andthe
NationalScienceFoundation,whichsupportedtheprogramsundergrantDMS
0932078000, andthe NationalSecurity Agency, which supported the workshops
throughgrantsH98230-12-1-0236/0256/0296/0298.
DavidEisenbud
SrikanthB.Iyengar
AnuragK.Singh
J.TobyStafford
MichelVandenBergh
ix